Weakly Primary Submodules over Non-commutative Rings

Abstract

Let R be an associative ring with nonzero identity and let M be a unitary left R −module. In this paper, we introduce the concept of weakly primary submodules of M and give some basic properties of these classes of submodules. Several results on weakly primary submodules over non-commutative rings are proved. We show that N is a weakly primary submodule of a left R −module M iff for every ideal P of R and for every submodule D of M with , 0 N PD   either) : (M N P  or N D . We also introduce the definitions of weakly primary compactly packed and maximal compactly packed modules. Then we study the relation between these modules and investigate the condition on a left R −module M that makes the concepts of primary compactly packed modules and weakly primary compactly packed modules equivalent. We also introduce the concept of weakly primary radical submodules and show that every Bezout module that satisfies the ascending chain condition on weakly primary radical submodules is weakly primary compactly packed module.

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Weakly Primary Submodules over Non-commutative Rings
ArwaEid Ashour1, Mohammad Hamoda2
1Department of Mathematics, The Islamic University of Gaza, Gaza, Palestine.
E-mail: arashour@iugaza.edu.ps
2Department of Mathematics, Al-Aqsa University, Gaza, Palestine.
E-mail: mamh_73@hotmail.com
Abstract
Let
R
be an associative ring with nonzero identity and let
M
be a unitary left
R
−module. In this paper,
we introduce the concept of weakly primary submodules of
M
and give some basic properties of these
classes of submodules. Several results on weakly primary submodules over non-commutative rings are
proved. We show that
N
is a weakly primary submodule of a left
R
−module
M
iff for every ideal
P
of
R
and for every submodule
D
of
M
with
,0 NPD 
either
):( MNP 
or
ND 
. We also
introduce the definitions of weakly primary compactly packed and maximal compactly packed modules.
Then we study the relation between these modules and investigate the condition on a left
R
−module
M
that makes the concepts of primary compactly packed modules and weakly primary compactly packed
modules equivalent. We also introduce the concept of weakly primary radical submodules and show that
every Bezout module that satisfies the ascending chain condition on weakly primary radical submodules
is weakly primary compactly packed module.
AMS Mathematics Subject Classification(2010): 13C05, 13C13, 13A15.
Keywords: Primary submodule; Weakly primary submodule; primary compactly packed module;
weakly primary compactly packed module; maximal compactly packed module; weakly primary radical
submodule.
1. Introduction
Throughout this paper, all rings are assumed to be associative not necessarily commutative with non-zero
identities, and all modules are unitary left modules. By "an ideal" we mean a 2-sided ideal.
Recently, extensive researches have been done on prime and primary ideals and submodules. The study
of prime submodules is one interesting topic in module theory. In particular, a number of papers
concerning prime submodules have been studied by various authors, see for example [9], [12], [15].
Weakly prime ideals in a commutative ring with nonzero identity have been introduced and studied by
Anderson and Smith in [2]. They defined a weakly prime ideal
P
over a commutative ring
R
with
identity as a proper ideal with the property that if whenever
Rba ,
with
,0 Pab
then either
Pa
or
Pb
. The structure of weakly primary ideals in a commutative ring has been studied by Atani and

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Farzalipour in [6]. They defined a weakly primary ideal
P
over a commutative ring
R
with identity as a
proper ideal with the property that if
,0 Pab
where
Rba ,
, then
Pa
or
n
bP
for some positive
integer
n
.The structure of weakly prime ideals over non-commutative rings has been studied by Hirano,
Poon, and Tsutsui in [10]. They defined a weakly prime ideal
P
over an associative ring
R
with identity
as a proper ideal with the property that if
PAB 0
implies either
PA 
or
PB 
for any ideals
BA,
of
R
. Recently, Ashour and Hamoda have been extended the concept of weakly primary ideals
over a commutative ring to non-commutative rings in [5]. They defined a right weakly primary ideal
P
over an associative ring
R
with identity as a proper ideal with the property that if whenever
BA,
are
ideals of
R
such that
PAB 0
, then
PA 
or Bn=bn: b BP for some n  . A proper
ideal
P
of R is called left weakly primary if whenever A, B are ideals of R such that
PAB 0
, then
B P or An=an: a AP for some n  . The ideal P is called weakly primary if it is both right
and left weakly primary.
The studying of prime submodules is extended in many ways, such as weakly prime submodules, primary
submodules, graded prime submodules, and n−absorbing submodules, see [7], [8], [17], [18].The
motivation of this paper is to continue the studying of the family of primary submodules, also to extend
the results of Atani and Frazalipour [7] and Smith [18] to the weakly primary submodules over non-
commutative rings. In fact, a number of results concerning weakly primary submodules over non
commutative rings are given.
We begin by reviewing the relevant definitions that are used in the sequel of this paper in Section 2.In
Section 3, we construct main results and theorems concerning weakly primary submodules over non-
commutative rings. We show that if N is a weakly primary submodule of a left R−module M with
0):( NMN
, then N is a primary submodule of M, (see Theorem 3.1.). The first main result of this
section is (Theorem3.3.). We show that N is a weakly primary submodule of M iff for every ideal Pof R
and for every submodule D of M with
,0 NPD 
either
):( MNP 
or
.ND 
One important
part of this section is the second main result(Theorem3.5.).We show that N is a weakly primary
submodule of M if for mM –N,
).:0():():( RmMNRmN
Finally, in Section 4, we introduce
the concepts of weakly primary compactly packed and maximal compactly packed modules and
investigate the relation between these concepts. Thus we show (Theorem 4.3.) that if M is a left
R−module with

0:)(  rRmMmMT
for some

.00  Rr
Then M is primary compactly
packed module if and only if M is weakly primary compactly packed module. Also, we give the main
result of this section (Theorem 4.7.). We show that if M is a Bezout module that satisfies the ascending
chain condition on weakly primary radical submodules, then M is a weakly primary compactly packed
module.
2. Preliminaries
We start by the following definition:
Definition 2.1. [14] Let R be an associative ring with identity, M be a left R−module, and N be a
submodule of M. The set (N : M) = {r R : rMN} is a left ideal of R called the left residual of N by M.
In particular, if m M, then
 
0:),0(  rmRrm
is called the left annihilator of m.
Similarly the right analogous for right residual and right annihilator can be defined for right R−modules.
Note that
),0( m
need not be a two-sided ideal of R. However if
),0( m
is a two-sided ideal of R, then
),0(),0( Rmm
.
Definition 2.2. [11] Let R be an associative ring with identity, let M be a left R−module.

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Then the set (0 : M) is a two sided ideal of R called the left annihilator of M.
Similarly the right analogous for the right annihilator can be defined for right R−modules.
In [9] Dauns defined the prime submodule over an associative ring with identity as follows:
Definition 2.3. Let R be an associative ring with identity and M be a left R−module. A
proper submodule N of M is called a prime submodule of M if rRmN (r R, m M), implies that either
m N or r (N : M).
Definition 2.4. [16] An associative ring R with identity is called a semi commutative ring
if ab = 0 implies aRb = 0 a, b R.
Definition 2.5. [13] An associative ring R with identity is called a local ring if it has a unique maximal
left (or right) ideal I of R denoted by (R,I).
In a commutative case we have the following definition:
Definition 2.6. [7] A proper submodule N of a module M over a commutative ring R is said to be weakly
prime submodule if whenever
,0 Nrm
for some r R, m M, then m N or rMN.
3. Weakly primary submodules
Our starting point is the following definitions:
Definition 3.1. Let R be an associative ring with identity, M be a left R−module, and N be a submodule
of M; then

NMRrMN rn :):(
for some positive integer n

is called the radical of a
submodule N over the ring R.
Definition 3.2. Let M be a left R−module. A proper submodule N of M is called a primary submodule of
M if whenever r R and m M with rRmN, then either m N or
).:( MNr 
Definition 3.3. Let M be a left R−module. A proper submodule N of M is called a weakly prime
submodule of M if whenever r R and m M with
,0 NrRm 
then either m N or r (N : M).
Definition 3.4. Let M be a left R−module. A proper submodule N of M is called a weakly primary
submodule of M if whenever r R and m M with
,0 NrRm 
then either m N or
).:( MNr 
Remarks 3.1.
(i) It’s clear that every primary submodule of a left R−module is a weakly primary submodule. However,
since 0 is always a weakly primary submodule (by definition), a weakly primary submodule does not
need to be primary.
(ii) We can prove directly from the definitions that every weakly prime submodule of a left R−module is
a weakly primary. However the converse is not true in general, since for example ifR = Z, the set of
integers, M = Z × Z and N = (4, 0)Z + (0, 1)Z, then N is a weakly primary submodule of M, however it is
not weakly prime submodule of M since
.)1,2(20 N
But neither 2M N nor (2, 1) N.
(iii) If N is a weakly primary submodule of a left R−module M, then (N : M) is not in general a weakly
primary ideal of R. For example, let M be the cyclic left Z−moduleZ/6Z. The zero module is a weakly
primary submodule of M, but (0 : M) = 6Z is not a weakly primary ideal of Z.

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(iv) If R is a commutative ring with identity. A proper submodule N is a weakly primary submodule of a
left R−module M iff whenever r R and m M with
,0 Nrm
then m N or
).:( MNr 
This is
clear by the equivalence rm N iff Rrm N.
As in the previous remark, we see that a weakly primary submodules need not be primary.
The following theorem gives the condition that makes the weakly primary submodule primary.
Theorem 3.1. Let N be a weakly primary submodule of a left R−module M. If
,0):( NMN
then N is
a primary submodule of M.
Proof. Let r R and m M with rRmN. If
,0rRm
then N is weakly primary submodule gives m N
or
).:( MNr 
So assume that rRm = 0. If
,0 rN
then x N such that
.0rx
Now
,)(0 NxmrRrRx 
so N is weakly primary submodule gives (m + x) N or
).:( MNr 
Thus m N or
).:( MNr 
Now we assume that rN = 0. If
,0):( mMN
then
):( MNk 
such that
.0km
So
.)(0 NRmkrkRm 
So m N or
).:()( MNkr 
Since
),:( MNk 
then we have m N or
).:( MNr 
So we can assume
that
.0):( mMN
Since
,0):( NMN
):( MNf 
and d N such that
.0fd
Then
,)()(0 NdmRfrfRd 
so (m+ d) N or
).:()( MNfr 
Thus m N or
):()( MNfr 
, so m N or
).:( MNr 
Now, the following result follows immediately from Theorem 3.1.
Corollary 3.2. Let N be a weakly primary submodule of a left R−module M. If N is not primary
submodule of M, then for any ideal P of R such that
):( MNP 
we have PN = 0. In particular
.0):( NMN
In a similar manner we can prove the following result:
Remark 3.2. If N is a weakly prime submodule of a left R−module M that is not prime, then forany ideal
P of R such that P (N : M), then PN = 0. In particular(N : M)N = 0.
Theorem 3.3. Let N be a proper submodule of a left R−module M. Then the following are equivalent:
(i) N is a weakly primary submodule of M.
(ii) For every ideal P of R and for every submodule D of M with
,0 NPD 
either
):( MNP 
or
D N.
Proof. (i)(ii) Suppose that N is a weakly primary submodule of M. If N is primary, then the result is
trivial. So assume that N is a weakly primary submodule of M that is not primary. Let
NPD 0
with
x D−N. We want to show that
).:( MNP 
Let r P. If
,0 rRx
since rRxN and N is weakly
primary, so
).:( MNr 
So assume that 0 = rRx. Now assume that
,0rD
say
0rt
for some t
D. Now
,0 NrRt 
then
).:( MNr 
If
,Nt 
then
).:( MNr 
If t N, then
,)(0 NxtrRrRt 
so (t + x) N or
).:( MNr 
Since
,Nx
then
).:( MNr 
So we

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can assume that rD = 0.Suppose that
,0Px
say
0ax
where a P. Now
.0 NaRx 
Then N is
weakly primary submodule gives
).:( MNa
As
,)(0 NRxaraRx 
we get
).:( MNr 
Therefore we can assume that Px = 0. Since
PpPD  ,0
and
D
t
1
such that
.0
1
t
p
Now
.0 1NpRt
As
.0):( NMN
(by Corollary3.2.) and
,)(0 11 NxpRpR tt 
we have two
cases:
Case(I).
):( MNp
and
.)( 1Nx
t
Since
,)()(0 11 NpRxRpr tt 
we obtain
):()( MNpr 
, so
).:( MNr 
Case (II).
):( MNp
and
.)( 1Nx
t
As
.0 1NpRt
We have
,
1N
t
so x N which is a contradiction.
Therefore
).:( MNr 
Thus
):( MNP 
.
(ii)(i) Assume that
NsRm 0
where s R and m M. Take P = Rs and D = Rm.
,0 NRsRmPD 
so either
):( MNP 
or D N. Thus
):( MNs
or m N.
Theorem 3.4. Let N be a proper submodule of a left R−module M. Then the following are equivalent:
(i) For ideal P of R and submodule D of M with 0PD N, either P (N : M) or D N.
(ii) N is a weakly prime submodule of M.
(iii) For mM –N, (N : Rm) = (N : M)(0 : Rm).
(iv) For mM –N, (N : Rm) = (N : M) or (N : Rm) = (0 : Rm).
Proof. (i)(ii) Suppose that
NsRm 0
where s R and m M. Take P = Rs and D = Rm. Then
,0 NPD 
so either P (N : M) or D N, hence either s (N : M) or m N. Thus N is weakly prime
submodule of M.
(ii)(i) Suppose that N is a weakly prime submodule of M. If N is prime then the result is clear. So we
can assume that N is a weakly prime submodule of M that is not prime.
Let
NPD 0
with x D−N. We need to show that
):( MNP 
. Let r P. If
,0 rRx
since
rRxN and N is weakly prime, so r (N : M). So assume that 0 = rRx. Now assume that
,0rD
say
0rt
for some t D. Now
.0 NrRt 
If
,Nt 
then r (N : M). If t N, then
,)(0 NxtrRrRt 
so (t + x) N or r (N : M). Since
,Nx
then r (N : M). So we can
assume that rD = 0. Suppose that
,0Px
say
0ax
where a P. Now
.0 NaRx 
ThenN is
weakly prime submodule gives
).:( MNa 
As
,)(0 NRxaraRx 
we get r (N : M), so P
(N : M). Therefore we can assume that Px = 0.Since
PpPD  ,0
and
D
t
1
such that
.0
1
t
p
As(N : M)N = 0 (by Remark 3.2.) and
,)(0 11 NxpRpR tt 
we have two cases:
Case(I). p (N : M) and
.)( 1Nx
t

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Since
,)()(0 11 NpRxRpr tt 
we obtain (r + p) (N : M),so r (N : M).
Case (II).
):( MNp 
and
.)( 1Nx
t
As
.0 1NpRt
We have
,
1N
t
so x N which is a contradiction.
Therefore r (N : M). Thus P (N : M).
(ii)(iii) If m M –N, then it is clear that K = (N : M)(0 : Rm) (N : Rm). Let
x (N : Rm). Then xRmN. If
,0 xRm
then x (N : M) since N is weakly prime submodule. If xRm
= 0, then x (0 : Rm).
(iii)(iv) Let m M−N, so that (N : Rm) = (N : M)(0 : Rm).Now (N : Rm), (N :M) and
(0 : Rm) are all ideals of R, that means (N : M)(0 : Rm) is an ideal of R and since the union of two
ideals of a ring is an ideal iff one of them is contained in the other, so we have (N : M) 
(0 : Rm) or (0 : Rm) (N : M), from which we get (N : Rm) = (N : M) or (N : Rm) = (0 : Rm).
(iv)(ii) Suppose that
NrRm 0
where r R and m M−N. Then r (N : Rm) and
).:0( Rmr
It
follows that r (N : M).
Theorem 3.5. Let N be a proper submodule of a left R−module M. Then the following are equivalent:
(i) N is a weakly primary submodule of M.
(ii) For mM –N,
).:0():():( RmMNRmN
Proof. (i)(ii) Assume that N is a weakly primary submodule of M and let r (N : Rm)
where m M –N. Thus rRmN. If
,0rRm
then N is weakly primary submodule gives
),:( MNr 
and hence
).:0():( RmMNr 
If rRm = 0, then r (0 : Rm) and hence
).:0():( RmMNr 
(ii)(i) Suppose that
NrRm 0
with r R and m M –N. Then r (N : Rm) and
).:0( Rmr
Now
):0():():( RmMNRmN
implies
).:( MNr 
Thus, N is weakly primary.
Proposition 3.6. Let N be a proper submodule of a left R−module M. Then the following are equivalent:
(i) For every ideal P of R and for every submodule D of M with PD N, either
):( MNP 
or D N.
(ii) N is a primary submodule of M.
(iii) For every left (or right) ideal P of R and for every submodule D of M with PD N, either
):( MNP 
or D N.
Proof. (i)(ii) Let r R and m M such that rRmN. It follows, since N is a submodule, that
(RrR)(Rm) N. Now, by (i), we get
):( MNRrR 
or RmN. If
):( MNRrR 
, then
):( MNr 
. If RmN, then m N. Therefore N is primary.

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(ii)(iii) Assume that PD N, for left (or right) ideal P R and submodule D M. If   ,then there
exists x D − N. For every t P, we have (tR)x = t(Rx) PD N which gives, by(ii),
).:( MNt 
(iii)(i) Obvious.
Now we introduce the following definition:
Definition 3.5. Let M be a left R−module, the subset T(M) of M is defined by
T(M) = {m M : rRm = 0 for some 0r R}
Note that if R is an integral domain, then it is easy to see that T(M) is a submodule of M.
Theorem 3.7. Let M be a left R−module with T(M) = 0. Then every weakly primary submodule of M is
primary.
Proof. Let N be a weakly primary submodule of M. Suppose that rRmN where rR, m M. If
NrRm 0
, then N is weakly primary submodule gives m Nor
).:( MNr 
If rRm = 0, then r =
0 or m = 0 (since T(M) = 0). Thus N is primary.
Proposition 3.8. Let M be a left module over a semi commutative local ring R with unique maximal left
ideal P. If PM = 0, then every proper submodule of M is weakly prime.
Proof. Let N be a proper submodule of M and let
NrRm 0
where r R andm M. Then
Nrm0
(because R is semi commutative).If r is a unit, then m N. If r is not a unit, then
rmPM = 0, a contradiction. Hence N is weakly prime.
Theorem 3.9. Let M1and M2 be left R−modules, M = M1M2 be a direct sum ofM1 and M2and let N
M1M2. Then the following are satisfied:
(i) If N = QM2 is a weakly primary submodule of M for some submodule Q of M1, then Q is aweakly
primary submodule of M1.
(ii) If N = M1Q is a weakly primary submodule of M for some submodule Q of M2,then Q is a weakly
primary submodule of M2.
Proof. (i) Let N = QM2 be a weakly primary submodule of M = M1M2. Let
QrRq 0
where r R, q M1 such that q Q, then (q, 0) QM2.
QqrR  )0,(0
M2. Since N = QM2 is a weakly primary submodule of M, a positive integer n
such that rn(M1M2) QM2.Hence rnM1Q for some positive integer n. So
).:( 1
M
Qr 
Therefore Q is weakly primary submodule of M1.
(ii) Proceed similar as in (i).
4. Weakly primary compactly packed modules
Primary compactly packed and primary finitely compactly packed modules have been introduced and
studied by Ashour in [3], [4]. In this section we study the concepts of weakly primary compactly packed
and maximal compactly packed modules.
Recall that a proper submodule N of a left R−module M is said to be maximal if there

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is no submodule K of M such that





.
Definition 4.1. [3] Let M be a left R−module. A submodule N of M is called primary compactly packed
submodule of M denoted by pcp−submodule of M if for each family{Pi : i I} of primary submodules of
M with N  iI Pi , N Pj for some j I.
M is called primary compactly packed module denoted by pcp−module if every submodule of M is a
pcp−submodule.
Now, we give the following definitions:
Definition 4.2. Let M be a left R−module.A submodule N of M is called weakly primary compactly
packed submodule of M denoted by wpcp−submodule of M if for each family
{Pi : i I} of weakly primary submodules of M with N iI Pi , N Pj for some j I.
M is called weakly primary compactly packed module denoted by wpcp−module if every submodule of
M is a wpcp−submodule.
Definition 4.3. Let M be a left R−module. A submodule N of M is called maximal compactly packed
submodule of M denoted by mcp−submodule of M if for each family{Pi : i I} of maximal submodules
of M with N  iI Pi , N Pj for some j I.
M is called maximal compactly packed module denoted by mcp−module if every submodule of M is a
mcp−submodule.
Now, we need the following Lemma:
Lemma 4.1. Every maximal submodule in a left R−module M is prime submodule.
Proof. Let K be a maximal submodule of M. Assume that r R and m M such that rRmK. Suppose
that
Km
. Then m + K is a nonzero element in M/K, which means that M/K is cyclicgenerated by m +
K. Hence for every x M, there exists t R such that x + K = t(m + K). It follows that x − tm K and
therefore
rx− rtmK. However, by the assumption rtmK and we conclude rxK and consequently r(K : M).
Remark 4.1. Clearly, every wpcp−module is pcp−module, and every pcp−module ismcp−module.
Theorem 4.2. Let M be a finitely generated left R−module. Then M is a mcp−module if and only if every
submodule N in M satisfies N + PiM for some i I where N iI Pi , Pi′s are weakly primary
submodules of M.
Proof. Let M be a finitely generated left R−module, and suppose that N is a submodule of M such that N
 iI Pi , Pi′s are weakly primary submodules of M. Foreach Pi , there exists a maximal submodule Mi
containing Pi . Then N  iIMi and so N Mi for some i I by hypothesis. Since PiMi , we have N +
PiMiM.
Conversely; let N be a submodule of M such that N  iIMi where each Mi is a maximal submodule of
M. Since every prime submodule is weakly prime submodule and every weakly prime submodule is
weakly primary submodule, so every maximal submoduleis weakly primary (by Lemma 4.1.), then N
+MiM for some i I. Therefore, since MiN +Mi M, then N+Mi = Mi , so N Mi for some i I.
The following theorem follows immediately from Theorem 3.7. and Remark 4.1.
Theorem 4.3. Let M be a left R−module with T(M) = 0. Then M is a pcp−module if and only if M is a
wpcp−module.

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Now, we give the following definition:
Definition 4.4. Let N be a submodule of a left R−module M. The intersection of all weakly primary
submodules containing N is called the weakly primary radical of N and is denoted by wprad(N). If there
is no weakly primary submodule containing N, then wprad(N) = M. In particular wprad(M) = M. We say
that a submodule N is a weakly primary radical submodule if wprad(N) = N.
The following result can be easily proved:
Proposition 4.4. Let N and L be submodules of a left R−module M. Then the following are hold:
(i) N  wprad(N).
(ii) wprad(wprad(N))= wprad(N), that is the weakly primary radical of N is a weakly primary radical
submodule.
(iii) If N L, then wprad(N) wprad(L).
(iv) wprad(N∩L) wprad(N)∩wprad(L).
Theorem 4.5. Let M be a left R−module. The following statements are equivalent:
(i) M is a wpcp−module.
(ii) For each proper submodule N of M, there exists m N such that wprad(N) = wprad(Rm).
(iii) For each proper submodule N of M, if {Ni : i I} is a family of submodules of Mand
N  iI Ni , then N wprad(Nj) for some j I.
(iv) For each proper submodule N of M, if {Ni : i I} is a family of weakly primaryradical submodules of
M and N iI Ni , then N Nj for some j I.
Proof. (i)(ii) Assume that M is a wpcp−module and let N be a proper submodule of M. It is clear that
wprad(Rm) wprad(N) for each m N. For the other inclusion, suppose that wprad(N) wprad(Rm) for
each m N. Then for each m N, there exists a weakly primary submodule Pm for which RmPm and N
 Pm . ButN =mNRm mN Pm , that is M is nota wpcp−module, which is a contradiction.
(ii)(iii) Let N be a proper submodule of M, and let {Ni : i I} be a family of submodules of Msuch that
N  iI Ni .By (ii), there exists m N such that wprad(N) = wprad(Rm).
Then m   iI Ni and hence m Nj for some j I. Hence N wprad(N) = wprad(Rm) wprad(Nj) for
some j I.
(iii)(iv) Let N be a proper submodule of M, and let {Ni : i I} be a family of weakly primary radical
submodules of M such that N  iI Ni . By (iii), there exists j I such that N wprad(Nj). Since Nj is
weakly primary radical submodule of M, thenN Nj .
(iv)(i) Let N be a proper submodule of M, and suppose that {Ni : i I} is a family of weakly primary
submodules of M such that N iI Ni .Since Ni is weakly primary submodule of M for each i I, Ni =
wprad(Ni) for each i I. Thus N  iI Ni =iIwprad(Ni). By (iv), there exists j I such that N
wprad(Nj) = Nj . Thus M is a wpcp−module.
Now we give the definition of the Bezout module over non commutative ring which is a generalization of
the definition of the Bezout module over commutative ring in [1].
Definition 4.5. A left R−module M is said to be a Bezout module, if every finitely generated submodule
of M is cyclic.

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Consider the following Lemma:
Lemma 4.6. Let M be a left R−module. If M satisfies the ascending chain condition on weakly primary
radical submodules, then any weakly primary radical submodule is the weakly primary radical of a
finitely generated submodule.
Proof. Assume that there exists a weakly primary radical submodule N which is not weakly primary
radical of a finitely generated submodules. Let m1N and N1 = wprad(m1R).Then N1N. So there exists
m2N−N1 .Let N2 = wprad(m1R+m2R). Then N1N2N. So that there exists m3N − N2 . Continuing
in this process, we will have an ascending chain of weakly primary radical submodules N1N2N3 · ·
·which is a contradiction.
Now, we are ready to prove the main result of this section.
Theorem 4.7. Let M be a Bezout module. If M satisfies the ascending chain condition on weakly primary
radical submodules, then M is wpcp−module.
Proof. Let N be a proper submodule of M. By Lemma 4.6., there exists a finitely generated submodule K
of M such that wprad(N) = wprad(K) and hence K is cyclic submoduleof M, because M is Bezout. It
follows by Theorem 4.5. that M is a wpcp−module.
Acknowledgements. The authors are grateful to Prof. M.H. Fahmy, and Prof. R.M. Salem, Math.
Dept. Fac. of Sci. Al-Azhar Univ. Egypt, for their useful comments.
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